The average absolute deviation, or simply average deviation of a data set is the average of the absolute deviations from a central point and is a summary statistic of statistical dispersion or variability. In this general form, the central point can be the mean, median, mode, or the result of another measure of central tendency. See below for distinctions where average is synonymous with "mean" and the central point is also the "mean".

The average absolute deviation of a set {x1, x2, ..., xn} is

The choice of measure of central tendency, , has a marked effect on the value of the average deviation. For example, for the data set {2, 2, 3, 4, 14}:

Measure of central tendency

Average absolute deviation

Mean = 5

Median = 3

Mode = 2

The average absolute deviation from the median is less than or equal to the average absolute deviation from the mean. In fact, the average absolute deviation from the median is always less than or equal to the average absolute deviation from any other fixed number.

For the normal distribution, the ratio of mean absolute deviation to standard deviation is . Thus if X is a normally distributed random variable with expected value 0 then, see Geary (1935):[1]

In other words, for a normal distribution, mean absolute deviation is about 0.8 times the standard deviation. However in-sample measurements deliver values of the ratio of mean average deviation / standard deviation for a given Gaussian sample n with the following bounds: , with a bias for small n.[2]

The mean absolute deviation (MAD), also referred to as the "mean deviation" or sometimes "average absolute deviation" is the mean of the data's absolute deviations about the data's mean: the average (absolute) distance from the mean. "Average absolute deviation" can refer to either this usage, or to the general form with respect to a specified central point (see above).

MAD has been proposed to be used in place of standard deviation since it corresponds better to real life.[3] Because the MAD is a simpler measure of variability than the standard deviation, it can be used as pedagogical tool to help motivate the standard deviation.[4][5]

This method's forecast accuracy is very closely related to the mean squared error (MSE) method which is just the average squared error of the forecasts. Although these methods are very closely related, MAD is more commonly used because it is both easier to compute (avoiding the need for squaring)[6] and easier to understand.[7]

More recently, the mean absolute deviation about mean is expressed as a covariance between a random variable and its under/over indicator functions;[8]

where

Dm is the expected value of the absolute deviation about mean,

"Cov" is the covariance between the random variable X and the over indicator function ().

and the over indicator function is defined as

Based on this representation new correlation coefficients are derived. These correlation coefficients ensure high stability of statistical inference when we deal with distributions that are not symmetric and for which the normal distribution is not an appropriate approximation. Moreover an easy and simple way for a semi decomposition of Pietra’s index of inequality is obtained.

For the example {2, 2, 3, 4, 14}: 3 is the median, so the absolute deviations from the median are {1, 1, 0, 1, 11} (reordered as {0, 1, 1, 1, 11}) with a median of 1, in this case unaffected by the value of the outlier 14, so the median absolute deviation (also called MAD) is 1.

The maximum absolute deviation about a point is the maximum of the absolute deviations of a sample from that point. While not strictly a measure of central tendency, the maximum absolute deviation can be found using the formula for the average absolute deviation as above with , where is the sample maximum. The maximum absolute deviation cannot be less than half the range.

The measures of statistical dispersion derived from absolute deviation characterize various measures of central tendency as minimizing dispersion: The median is the measure of central tendency most associated with the absolute deviation. Some location parameters can be compared as follows:

trimmed L∞ norm statistics: for example, the midhinge (average of first and third quartiles) which minimizes the median absolute deviation of the whole distribution, also minimizes the maximum absolute deviation of the distribution after the top and bottom 25% have been trimmed off.

The mean absolute deviation of a sample is a biased estimator of the mean absolute deviation of the population. In order for the absolute deviation to be an unbiased estimator, the expected value (average) of all the sample absolute deviations must equal the population absolute deviation. However, it does not. For the population 1,2,3 both the population absolute deviation about the median and the population absolute deviation about the mean are 2/3. The average of all the sample absolute deviations about the mean of size 3 that can be drawn from the population is 44/81, while the average of all the sample absolute deviations about the median is 4/9. Therefore the absolute deviation is a biased estimator.
However, this argument is based on the notion of mean-unbiasedness. Each measure of location has its own form of unbiasedness (see entry on biased estimator. The relevant form of unbiasedness here is median unbiasedness.